Assuming f E C3(R3) and g E C23) in C2 (R3 x (0, oo). , show that u E u(x, t)- ot 4Tc2t X = (2.1,...
Sketch the product of these, viz., g(t)=u(t-3) u(-t-4) f g(t)dt = u(t-3) u(-t-4)dt Evaluate t=-oo t=-oo Evaluate ) f u(C)dt and (i) f u()dt t-500 t=-oo
Sketch the product of these, viz., g(t)=u(t-3) u(-t-4) f g(t)dt = u(t-3) u(-t-4)dt Evaluate t=-oo t=-oo Evaluate ) f u(C)dt and (i) f u()dt t-500 t=-oo
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
(b) Let E = {(1, C2, C3} be the standard basis for R3, B = {bų, b2, b3} be a basis for a vector space U, and S: R3 → U be a linear transformation with the property that S(X1, X2, X3) (x2 + x3)b1 + (x1 + 3x2 + 3x3)b2 + (-3X1 - 5x2 - 4x3)b3. Find the matrix F for S relative to E and B. INSTRUCTIONS: 1. Use the green arrows next to the answer spaces below...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
C3.)Let f(z,0) (1/θ)2(1-0)/0, 0 < x < 1,0 < θ < oo. . Find the maximurn likelihood estimator of θ. Show that the maximurn likelihood estimator is unbiased to θ
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
4. Let u be the solution of the Burgers quaslilinear 1st order PDE u, + uu,-0, a(0,x) = g(x) E C2(R2) and suppose that lIgl(R) oo and that u E C"(-T,T) × R). Prove that 2. if g has compact support, then so does u(t,.) for allt E (-T,T); 3. if g 20and has compact support, then 4. if g 2 0 and has compact support, then f(u(t,x))dx= | f(g(x))dx for all f e C([0, oo)).
Show that
is the general solution for the PDE
f(x, t) = C(2.1 + 3t) + c2(2.1 – 3t) 4ft = 91.1
d1=8
d2=9
lu for Find the solution u(x,t) for the l-D wave equation-=- Qx2 25 at2 (a) oo < x < oo with initial conditions u(x,0)-A(x) , where A(x) Is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. d2+5 di+10 di+15dı+20 (b) Check for the wave equation in (a) that if (x...
4. Solve the initial, boundary value problem by the Fourier integral method te = kurz, 0<x<oo, t>0, us(0,t) = 0, u(x, t) bounded as T-100 0S$ 0, >4 f(x)-( 4 u(z,0)=f(x),
4. Solve the initial, boundary value problem by the Fourier integral method te = kurz, 04 f(x)-( 4 u(z,0)=f(x),